The one-dimensional stochastic Keller--Segel model with time-homogeneous spatial Wiener processes

TL;DR

This paper investigates a stochastic version of the classical Keller--Segel chemotaxis model perturbed by spatial Wiener processes, proving the existence of solutions under random influences, which models inherent biological fluctuations.

Contribution

It introduces a stochastic Keller--Segel model with spatial Wiener noise and proves the existence of martingale solutions, extending deterministic chemotaxis models to include randomness.

Findings

Existence of martingale solutions for the stochastic system.

Extension of classical chemotaxis models to stochastic settings.

Mathematical framework for analyzing randomness in chemotaxis models.

Abstract

Chemotaxis is a fundamental mechanism of cells and organisms, which is responsible for attracting microbes to food, embryonic cells into developing tissues, or immune cells to infection sites. Mathematically chemotaxis is described by the Patlak--Keller--Segel model. This macroscopic system of equations is derived from the microscopic model when limiting behaviour is studied. However, on taking the limit and passing from the microscopic equations to the macroscopic equations, fluctuations are neglected. Perturbing the system by a Gaussian random field restitutes the inherent randomness of the system. This gives us the motivation to study the classical Patlak--Keller--Segel system perturbed by random processes. We study a stochastic version of the classical Patlak--Keller--Segel system under homogeneous Neumann boundary conditions on an interval $\mathcal{O}=[0,1]$. In particular, let $\mathcal{W}_1$, $\mathcal{W}_2$ be two time-homogeneous spatial Wiener processes over a filtered probability space $\mathfrak{A}$. Let $u$ and $v$ denote the cell density and concentration of the chemical signal. We investigate the coupled system \begin{align*} & d {u} - ( r_u\Delta u- \chi {\rm div }( u\nabla v) )\, dt =u\circ d\mathcal{W}_1, \\ & d{v} -(r_v \Delta v -\alpha v)\, dt = \beta u \, dt+ v\circ d\mathcal{W}_2, \end{align*} with initial conditions $(u(0),v(0))=(u_0,v_0)$. The positive terms $r_u$ and $r_v$ are the diffusivity of the cells and chemoattractant, respectively, the positive value $\chi$ is the chemotactic sensitivity, $\alpha\ge0$ is the so-called damping constant. The noise is interpreted in the Stratonovich sense. Given $T>0$, we will prove the existence of a martingale solution on $[0,T]$.